Tagged Questions

I am aware that the formulas for complex sentences are ID and D,I (I=independent and D=dependent), but I have a sentence that has an I and D but adding a subordinate conjunction sounds weird.
"He woke ...

I am reading this The Elements of Style book by Strunk and White, I am confused about rule number 11.
A participial phrase at the beginning of a sentence must refer to
the grammatical subject
There ...